A rate of change tells how much an output changes for each 1-unit change in the input: the change in y divided by the change in x. If a phone plan costs 15 dollars more for every extra hour, its rate of change is 15 dollars per hour. For a straight line, the rate of change equals the slope.
In Spanish
tasa de cambio
Also called
average rate of change, change in y over change in x, Δy/Δx
In the table every hour adds 15 dollars, so the rate never changes. On the curve the rate changes; between x = 1 and x = 3 its average rate of change is 82 = 4.
Formal definition and notation
The average rate of change of a functionf on the interval from x1 to x2, with x1 ≠ x2, is f(x2) − f(x1)x2 − x1. It is measured in output units per input unit. A function has a constant rate of change on every interval exactly when it is linear; for other functions the average rate of change depends on the interval chosen.
Symbol
Read it as
Meaning
Δ yΔ x
delta y over delta x
change in output divided by change in input
f(b) − f(a)b − a
f of b minus f of a, over b minus a
the average rate of change from x = a to x = b
m
m
the slope, which is the constant rate of change of a line
Key facts
In a table, find the rate of change by dividing the change in the output column by the change in the input column. If the inputs jump by 2, divide by 2.
A constant rate of change means the graph is a straight line (linear). The rate is the same between any two points.
A curve has a changing rate of change. Its average rate over an interval is the slope of the segment joining the two endpoints.
The units matter: a rate of change of −15 gallons per minute means a tank is losing 15 gallons each minute.
rate of change = y2 − y1x2 − x1 = change in outputchange in input
(x1, y1)
the starting input and output
(x2, y2)
the ending input and output
units
output units per input unit, such as miles per hour or dollars per month
A positive rate means the output increases as the input increases; a negative rate means it decreases; a rate of 0 means no change.
Try it: change in y over change in x
Interactive
Drag the two points. The readout shows the change in y, the change in x and their quotient, the rate of change of the line between them.
This interactive needs JavaScript. The worked examples below show the same ideas step by step.
Rate of Change examples, step by step
Example 1: Rate of change from a table
Warm-up
A table shows x = 2, y = 10 and x = 6, y = 22 for a linear function. What is the rate of change?
Change in output: 22 − 10 = 12.
Change in input: 6 − 2 = 4.
Divide: 124 = 3.
Answer: The rate of change is 3: y goes up 3 for each 1 that x goes up.
Example 2: Average rate of change of a curve
Core
Find the average rate of change of f(x) = x2 + 1 from x = 1 to x = 4.
Outputs: f(1) = 1 + 1 = 2 and f(4) = 16 + 1 = 17.
Change in output: 17 − 2 = 15.
Change in input: 4 − 1 = 3.
Average rate of change: 153 = 5.
Answer: The average rate of change is 5 on the interval from 1 to 4.
Example 3: A negative rate with units
Test-style
A water tank holds 500 gallons. After 12 minutes of draining it holds 320 gallons. What is the average rate of change of the water in the tank, and what does it mean?
Points (minutes, gallons): (0, 500) and (12, 320).
Change in gallons: 320 − 500 = −180.
Change in time: 12 − 0 = 12 minutes.
Rate: −18012 = −15 gallons per minute. The negative sign means the amount is going down.
Answer: The rate of change is −15 gallons per minute: the tank loses 15 gallons each minute on average.
Common mistakes
✗ MistakeFrom (2, 10) to (6, 22), the rate of change is 22 − 10 = 12.
✓ InsteadDivide by the change in x: 124 = 3.
A rate of change is per 1 unit of input. When the inputs jump by more than 1, the change in output must be divided by that jump.
✗ MistakeDraining from 500 to 320 gallons in 12 minutes is a rate of 15 gallons per minute.
✓ InsteadThe rate is −15 gallons per minute.
Subtract ending minus starting: 320 − 500 is negative. The sign tells whether the quantity grows or shrinks, and tests often ask exactly that.
✗ Mistakef(x) = x2 has a rate of change of 4 everywhere, because from 1 to 3 it is 4.
✓ InsteadFrom 0 to 1 it is 1, and from 3 to 5 it is 8: the rate changes.
Only linear functions have a constant rate of change. For a curve, always name the interval.
Slope is the steepness of a line on a graph. Rate of change is the same number described in context with units, such as 15 dollars per hour, and it also applies to tables and curves.
A rate compares two amounts, such as 150 miles in 3 hours. A rate of change compares how one quantity changes as another changes, such as speed rising by 4 miles per hour every second.
Rate of change is the amount of change per unit of input (−15 gallons per minute). Percent change compares the change with the starting amount: from 500 to 320 is a 36 percent decrease.
Quick check: 3 questions
Question 1 of 3
A car’s distance from home is 40 miles at 1 p.m. and 190 miles at 4 p.m. What is its average rate of change?
Show answer and explanation
Answer: B.190 − 404 − 1 = 1503 = 50 miles per hour. 150 forgets to divide by the time, 47.5 divides 190 by 4, and 0.02 divides time by distance.
Question 2 of 3
Which situation has a negative rate of change?
Show answer and explanation
Answer: B. The candle’s height decreases as time increases, so its rate of change is −0.5 inches per hour. The savings and the plant have positive rates, and the constant fare has a rate of 0.
Question 3 of 3
What is the average rate of change of g(x) = 2x2 from x = 1 to x = 3?
Show answer and explanation
Answer: D.g(3) = 18 and g(1) = 2, so 18 − 23 − 1 = 162 = 8. 16 forgets to divide by 2, 18 is just g(3), and 4 is half the right answer.
Rate of Change: frequently asked questions
Is rate of change the same as slope?
For a straight line, yes: the rate of change is the slope, and it is the same between any two points. The words are used differently, though. Slope describes the graph, while rate of change describes the situation with units, such as 15 dollars per hour. For a curve, the rate of change is not constant, so you find an average rate of change over a chosen interval.
How do you find the rate of change from a table?
Pick two rows. Subtract the outputs, subtract the inputs in the same order, and divide the first difference by the second. If the table is linear, any two rows give the same answer. If different pairs of rows give different answers, the relationship is not linear, and each answer is an average rate of change for that interval.
What does a rate of change of zero mean?
It means the output does not change when the input changes. On a graph it is a horizontal line, such as a parking fee of 10 dollars no matter how many hours you stay. For a curve, an average rate of change of 0 on an interval means the output ended where it started, even if it went up and down in between.
Updated November 2026·Reviewed by the Effortless Math team·Standards: CCSS 8.F.B.4, CCSS HSF-IF.B.6
Cite this entry
Effortless Math Team. “Rate of Change.” Effortless Math Dictionary. Updated November 2026. https://www.effortlessmath.com/math-dictionary/rate-of-change/
Flashcard
TermRate of ChangeΔ yΔ x
DefinitionHow much the output changes per 1 unit of input. From (2, 10) to (6, 22) the rate of change is 12 ÷ 4 = 3.
SlopeSlope is a number that measures how steep a line is: the change in y divided by the change in x between any two points on the line, often called rise over run. A positive slope rises from left to right, a negative slope falls, and a horizontal line has slope 0.Algebra · Grade 8, Algebra 1Average Rate of ChangeThe average rate of change of a function over an interval is the change in output divided by the change in input: f(b) − f(a)b − a. It tells how fast the function changes, on average, from x = a to x = b. On a graph it is the slope of the line through the two endpoints.Functions · Algebra 1, Algebra 2, PrecalculusFunctionA function is a rule that gives exactly one output for each input. If you put in one x-value, a function gives back one y-value. In f(x) = 2x + 1, the input 3 gives the output 7. Two inputs may share an output, but one input can never have two different outputs.Functions · Grade 8, Algebra 1, Algebra 2Linear EquationA linear equation is an equation whose graph is a straight line. In one variable, it usually has a variable to the first power, like 3x + 2 = 14. In two variables, it can be written in forms like y = mx + b, where m is the slope. No variable is squared, multiplied by another variable or in a denominator.Algebra · Grade 7, Grade 8, Algebra 1AverageAn average is one number that stands for the typical size of the values in a data set. In everyday math, average means the mean: add all the values, then divide by how many values there are. In statistics, the median and the mode are also kinds of average.Statistics · Grade 5, Grade 6, Grade 7ProportionA proportion is an equation that says two ratios are equal, such as 12 = 36. It is read 1 is to 2 as 3 is to 6. In a true proportion the cross products are equal, which lets you find a missing value. Proportions are often used to solve scale, percent, and rate problems.Ratios and Proportions · Grade 6, Grade 7, Grade 8Constant of ProportionalityThe constant of proportionality is the constant ratio between two proportional quantities, k = yx. It is the k in y = kx and stays the same for every pair of values, which makes the graph a straight line through the origin. If 3 notebooks cost 9 dollars, k = 3 dollars per notebook.Ratios and Proportions · Grade 7, Grade 8, Algebra 1OriginThe origin is the point where the x-axis and the y-axis cross on the coordinate plane. Its coordinates are (0, 0), and it is labeled O. Every ordered pair tells how far to move from the origin: (4, 3) means 4 units right, then 3 units up.Geometry · Grade 5, Grade 6, Grade 8Linear FunctionA linear function is a function whose graph is a straight line. You can write it as f(x) = mx + b, where m is the slope, or constant rate of change, and b is the y-intercept. Each time x increases by 1, the output changes by the same amount, m.Functions · Grade 8, Algebra 1NumberA number is an amount or value used to count, measure, order and compare. We write numbers with digits (7), words (seven) or other symbols, but the number itself is the amount: 7 dots, the word seven and the numeral 7 all show the same number. Numbers include whole numbers, fractions, decimals and negatives.Number System · Kindergarten, Grade 1, Grade 2RateA rate is a ratio that compares two quantities measured in different units, such as 150 miles in 3 hours or 12 dollars for 4 pounds. The word “per” signals a rate. Dividing gives the unit rate, the amount for 1 unit: 150 miles in 3 hours is 50 miles per hour.Ratios and Proportions · Grade 6, Grade 7MileA mile (mi) is a US customary unit of length for long distances, equal to 5,280 feet or 1,760 yards. One mile is about 1.61 kilometers, so it is longer than a kilometer. Road distances and speed limits in the United States use miles and miles per hour.Measurement · Grade 4, Grade 5, Grade 6Percent ChangePercent change measures how much a value grew or shrank relative to where it started: the difference divided by the original amount, written as a percent. From 50 to 60 is a 20% increase; from 80 to 60 is a 25% decrease. The original value is always the base, which is the step people get wrong.Ratios and Proportions · Grade 7, Grade 8, Algebra 1Percent DecreasePercent decrease is how much a quantity went down, written as a percent of its original value. Subtract the new value from the original, divide by the original, and multiply by 100. A price that drops from 80 dollars to 60 dollars has a decrease of 20 dollars, and 2080 = 25%.Ratios and Proportions · Grade 7, Grade 8, Algebra 1DifferenceA difference is the result of subtracting one number from another. In 12 − 7 = 5, the difference is 5. It tells how much bigger one number is than another, or how far apart they are on a number line. “The difference of a and b” means a − b, in that order.Operations · Grade 1, Grade 2, Grade 3RatioA ratio compares two quantities by division. Ratios can be written with words, a colon, or a fraction: 3 boys to 5 girls is 3 to 5, 3 : 5, or 35. Order matters, and a ratio can compare part to part or part to whole. Multiplying both numbers by the same amount gives an equivalent ratio.Ratios and Proportions · Grade 6, Grade 7, Grade 8Ordered PairAn ordered pair is two numbers written to locate a point on the coordinate plane. It is written as (x, y): the x-value comes first, and the y-value comes second. From the origin, move x units across, then y units up or down. The order matters: (2, 5) and (5, 2) are different points.Geometry · Grade 5, Grade 6, Grade 7DerivativeA derivative measures how fast a function’s output changes at one exact input: its instantaneous rate of change. On a graph it is the slope of the tangent line at that point. The derivative of f(x) = x2 is f′(x) = 2x, so at x = 3 the curve is rising with slope 6.Calculus · CalculusUnit RateA unit rate is a rate that compares a quantity to one unit of another quantity, such as 50 miles per hour or 4 dollars per notebook. To find it, divide the first quantity by the second: 150 miles in 3 hours is 150 ÷ 3 = 50 miles per hour. The word per means for each one.Ratios and Proportions · Grade 6, Grade 7, Grade 8